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Lesson 5 of 6
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- Year 11
Signs of a solution
I can deduce why a change of sign may indicate a solution.
Lesson details
Key learning points
- If the value is a solution, then substituting it into the equation should mean the equation evaluates to 0
- Since the solution is an approximation, it is unlikely to be exactly 0
- By substituting the upper and lower bonds of the estimated value into the equation, you can tell if the estimate is good
- If the two values produce one positive and one negative value, this implies the graph would cross the x axis
- The x intercept would be a solution to the equation
Keywords
Iteration - Iteration is the repeated application of a function or process in which the output of each iteration is used as the input for the next iteration.
Lower bound - The lower bound for a rounded number is the smallest value that the number could have taken prior to being rounded.
Upper bound - The upper bound for a rounded number is the smallest value that would round up to the next rounded value.
Common misconception
A change in sign always indicates a solution. No change in sign means no solution.
A change in sign only indicates a solution if the equation is equal to zero and if the graph of the equation crosses the $$x$$ axis between those points. Where there is a repeated root or two solutions between them there may be no change in sign.
Using graphing software to look at the shape of different graphs will help with misconceptions. LC2 is a good chance for pupils to practise iteration skills and now test their answers to a suitable degree of accuracy.
Teacher tip
Licence
This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).
Lesson video
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Prior knowledge starter quiz
6 Questions
Q1.A value has been rounded to 2.8 to 2 significant figures. What is the lower bound of this value?
A value has been rounded to 2.8 to 2 significant figures. What is the lower bound of this value?
2
2.5
2.7
Correct answer: 2.75
2.75
Q2.A value has been rounded to 2.8 to 2 significant figures. What is the upper bound of this value?
A value has been rounded to 2.8 to 2 significant figures. What is the upper bound of this value?
Correct Answer: 2.85
2.85
Q3.What is the value of $$x^3-2x+5$$ when $$x=2$$?
What is the value of $$x^3-2x+5$$ when $$x=2$$?
Correct Answer: 9
9
Q4.What is the value of $$x^3-2x+5$$ when $$x=-2$$?
What is the value of $$x^3-2x+5$$ when $$x=-2$$?
Correct Answer: 1
1
Q5.Match the letter representing each graph to the equation of the graph.
Match the letter representing each graph to the equation of the graph.
Correct Answer:A,$$y=x^2$$
A -Â $$y=x^2$$
Correct Answer:B,$$y=x^3$$
B -Â $$y=x^3$$
Correct Answer:C,$$y=-x^2$$
C -Â $$y=-x^2$$
Correct Answer:D,$$y=\frac{1}{x}$$
D -Â $$y=\frac{1}{x}$$
Q6.Using the iterative formula $$x_{n+1}=\sqrt[3]{4(x_n)^2-1}$$ and $$x_0=5$$ there is a solution to $$x^3-4x^2+1=0$$ correct to 2 significant figures when $$x=$$ .
Using the iterative formula $$x_{n+1}=\sqrt[3]{4(x_n)^2-1}$$ and $$x_0=5$$ there is a solution to $$x^3-4x^2+1=0$$ correct to 2 significant figures when $$x=$$ .
Correct Answer: 3.9
3.9
Assessment exit quiz
6 Questions
Q1.Using the table of values shown for the equation $$y=x^3-4x+2$$ which of these statements are definitely true?
Using the table of values shown for the equation $$y=x^3-4x+2$$ which of these statements are definitely true?
There is a solution to the equation $$x^3-4x+2=0$$ between $$x=0$$ and $$x=0.5$$
Correct answer: There is a solution to the equation $$x^3-4x+2=0$$ between $$x=0.5$$ and $$x=1$$
There is a solution to the equation $$x^3-4x+2=0$$ between $$x=0.5$$ and $$x=1$$
Correct answer: There is a solution to the equation $$x^3-4x+2=0$$ between $$x=1$$ and $$x=2$$
There is a solution to the equation $$x^3-4x+2=0$$ between $$x=1$$ and $$x=2$$
There is a solution to the equation $$x^3-4x+2=0$$ between $$x=1$$ and $$x=1.2$$
Correct answer: There is a solution to the equation $$x^3-4x+2=0$$ between $$x=1.6$$ and $$x=2$$
There is a solution to the equation $$x^3-4x+2=0$$ between $$x=1.6$$ and $$x=2$$
Q2.Between which sets of values is there definitely a solution to $$x^3-5x-1=0$$ ?
Between which sets of values is there definitely a solution to $$x^3-5x-1=0$$ ?
Correct answer: Between $$x=-3$$ and $$x=-2$$
Between $$x=-3$$ and $$x=-2$$
Between $$x=-2.3$$ and $$x=-2.2$$
Between $$x=-0.4$$ and $$x=-0.5$$
Correct answer: Between $$x=2.3$$ and $$x=2.4$$
Between $$x=2.3$$ and $$x=2.4$$
Q3.When might a change in sign not indicate there is a solution between two values?
When might a change in sign not indicate there is a solution between two values?
Correct answer: When the equation being solved is not equal to zero.
When the equation being solved is not equal to zero.
When one input is negative but the other is positive.
When the values are really far apart.
Correct answer: When the graph has an asymptote and doesn't cross the axis between those points.
When the graph has an asymptote and doesn't cross the axis between those points.
Q4.The image shows Jacob's workings to test his approximate solution to the equation $$x^3-2x=0$$. What do you think his solution was?
The image shows Jacob's workings to test his approximate solution to the equation $$x^3-2x=0$$. What do you think his solution was?
$$x=1.3$$
$$x=1.35$$
$$x=1.355$$
Correct answer: $$x=1.4$$
$$x=1.4$$
$$x=1.45$$
Q5.To see if $$x=-1.4$$ is a solution to the equation $$x^3-2x=0$$ to 2 significant figures which values should we test to see if there is a change in sign?
To see if $$x=-1.4$$ is a solution to the equation $$x^3-2x=0$$ to 2 significant figures which values should we test to see if there is a change in sign?
$$x=-1$$ and $$x=-2$$
$$x=-1.3$$ and $$x=-1.5$$
Correct answer: $$x=-1.35$$ and $$x=-1.45$$
$$x=-1.35$$ and $$x=-1.45$$
$$x=-1.4$$ and $$x=-1.45$$
Q6.Which of these are solutions to $$-x^2+3x-1=0$$ correct to the stated degree of accuracy?
Which of these are solutions to $$-x^2+3x-1=0$$ correct to the stated degree of accuracy?
$$x=2$$ to 1 significant figure
Correct answer: $$x= 3$$ to 1 significant figure
$$x= 3$$ to 1 significant figure
Correct answer: $$x=0.38$$ to 2 significant figures
$$x=0.38$$ to 2 significant figures
$$x=2.5$$ to 1 decimal place
